Kevin, we have consonants like n, t, d, b, p that can be both 'soft' and 'hard'. Before и the will sounds soft. Before ы they will sound hard.
Also we have consonants like ч that can only be soft, and consonants like ш, that can only be hard.
- after 'soft' consonants.
путь - на пути
лень - лени
Юля - Юли
борщ - борщи
грач - грачи
Ирландия - Ирландии
Up to this momnet it is simple: idea is that, н and т just don't change. They were soft in other case forms, they remain soft in this one. Ч and щ and /j/ (as in boy /boj/) hidden in Ирландия are just always soft.
- after ш and ж
(just orthography, if Ы was written there it would sound the same)
- after к, г and х.
They DO become soft when -и is added!
For some reason, ы sound after г/к/х sounds rude (and funny ) for Russian ear. It only occurs in a few foreign words.